Motion Under Constant Power

When power is held constant, the energy delivered grows linearly with time, $W=Pt$, and by the work-energy theorem this equals the kinetic energy gained from rest, $Pt=\frac{1}{2}m{v}^2$. Solving for speed gives $v=\sqrt{2Pt/m}$, which shows that under constant power the speed rises in proportion to the square root of time, not linearly. Substituting $P=16\text{ W}$, $t=4\text{ s}$, and $m=2\text{ kg}$ gives $v=\sqrt{(2\times 16\times 4)/2}=\sqrt{64}=8\text{ m/s}$. The option 16 m/s reports $v^2$ without the square root. The option 4 m/s wrongly assumes speed grows linearly with time. The option 32 m/s omits the division by mass. The square-root dependence on time carries clear physical meaning: as the object speeds up, the fixed power must be shared with an ever-larger velocity, so the driving force and hence the acceleration steadily diminish. This contrasts sharply with constant-force motion, where the speed grows linearly with time and the acceleration remains fixed throughout. This is the NCERT result for one-dimensional motion under constant power. As a check, the energy delivered is $16\times 4=64\text{ J}$, matching $\frac{1}{2}(2)(8{)}^2=64\text{ J}$ exactly.